Accurate analytical and numerical modeling of multiscale systems is a daunting task. The need to properly resolve spatial and temporal scales spanning multiple orders of magnitude pushes the limits of both our theoretical models as well as our computational capabilities. Rigorous upscaling techniques enable efficient computation while bounding/tracking errors and helping to make informed cost-accuracy tradeoffs. The biggest challenges arise when the applicability conditions of upscaled models break down. Here, we present a non-intrusive two-way (iterative bottom-up top-down) coupled hybrid model, applied to thermal runaway in battery packs, that combines fine-scale and upscaled equations in the same numerical simulation to achieve predictive accuracy while limiting computational costs. First, we develop two methods with different orders of accuracy to enforce continuity at the coupling boundary. Then, we derive weak (i.e., variational) formulations of the fine-scale and upscaled governing equations for finite element (FE) discretization and numerical implementation in FEniCS. We demonstrate that hybrid simulations can accurately predict the average temperature fields within error bounds determined a priori by homogenization theory. Finally, we demonstrate the computational efficiency of the hybrid algorithm against fine-scale simulations.

翻译：多尺度系统的精确解析与数值建模是一项艰巨的任务。为准确解析跨越多个数量级的空间与时间尺度，既对我们的理论模型提出挑战，也考验计算能力。严格的升尺度技术能够在约束/追踪误差的同时实现高效计算，并辅助权衡成本与精度。当升尺度模型的适用条件失效时，最大挑战随之出现。本文提出一种非侵入式双向（迭代自底向上-自顶向下）耦合混合模型，应用于电池组热失控场景。该模型在统一数值模拟中联合使用细尺度与升尺度方程，在限制计算成本的同时实现预测精度。首先，我们开发了两种不同精度阶次的方法以强制执行耦合边界处的连续性。随后，推导细尺度与升尺度控制方程的弱形式（即变分形式），用于有限元（FE）离散化及在FEniCS中的数值实现。我们证明，混合模拟能够准确预测平均温度场，且误差落在均匀化理论预先确定的范围内。最后，通过对比细尺度模拟，验证了混合算法的计算效率。